After chairing an emergency meeting of Egypt's top military and intelligence brass, Mursi said he had ordered the army to take "full control of the Sinai", which has become hotbed of 'jihadists' and other lawless elements released recently under a mass amnesty.
The 1997 Peace Treaty between Israel and Egypt restricts number of Egyptian troops in Sinai.
Egyptian security sources said the five gunmen dressed up in Bedouin attire drove up to a border checkpoint between Egypt and the West Bank and opened indiscriminate fire,leaving 16 border guards dead.
But in a dramatic turn of events, the five gunmen involved in the attack were reported killed hours later near the Israeli border, an Israeli army spokesman said.
Israeli tanks and helicopters attacked and shot dead the 'jihadists' as they attempted to cross into Israel. The spokesman said two armoured vehicles were seized one of which exploded.
"The bodies of the five gunmen have been found by the Israeli army," the spokesman said.
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The Israeli spokesman said that swift action by their forces came as they were ready for the attackers after being tipped off Shin Bet (Israeli intelligence agency).
He said the gunmen were "members of the global jihad based in Sinai, which has become a hotbed of international terrorism".
The official MENA news agency also described the attackers as 'jihadists' from the Hamas-controlled Gaza strip.
Eyewitnesses said the attack came as the soldiers at the checkpoint were breaking their Ramadan fast. They said the soldiers were caught totally off guard and had no chance to retaliate.
Hundreds of Islamists were released under a general amnesty just two weeks ago in a first major decision by the new President.